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If a and b are non-negative integers, ab=? 1) 9^a=3^b 2) 5^a=2^b

ANS: From the original condition, there are 2 variables. Hence, there is a high chance that C is the correct answer. However, since this is an integer question, one of key questions, we should apply the common mistake type 4(A). From con 1), we get 2a=b. The answer is not unique and the condition is not sufficient. In case of con 2), we get a=b=0. The answer is unique and the condition is sufficient. Hence, the correct answer is B.
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If x, y are positive integers, what is the remainder when 2^(8x+y) is divided by 5? 1) x=1 2) y=2

ANS: From the original condition, there are 2 variables (x and y). In order to match the number of variables and the number of equations, we need 2 equations. Hence, there is a high chance that C is the correct answer. However, even though C is indeed the correct answer, since it is an integer question, one of the key questions, we have to apply the common mistake type 4(A). From the condition 2), the remainder of 2^(8x+y) is always 4. Hence, it is sufficient and the correct answer is B.
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Is x+y>0? 1) x>|y| 2) y<0 ANS: In case of con 1), from x>|y|≥-y, we get x>-y, x+y>0. The answer is yes and the condition is sufficient. Hence, the correct answer is A
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If x, y are positive integers, what is the unit digit of 2^(4x+2)+y? 1) x=1 2) y=2

ANS: In the original condition, there are 2 variables (x and y). Hence, there is a high chance that C is the correct answer. Using 1) & 2), C is the correct answer. However, since this is an integer question, we can apply the common mistake type 4(A). From con 2), the unit digit of 2^4x+2 is always 5. Hence, we only have to know y, and the condition is sufficient. The correct answer, thus, is B.
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ANS: Since there are 2 variables in the original condition, there is a high chance that C is the correct answer. Condition 2) becomes the answer too easily because from 1/x>1/4, we get 0<x<4. The condition is sufficient. However, the condition is rather too hard. If we apply the common mistake type 4(B), the answer becomes D.
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ANS: There are 2 variables in the original condition. In order to match the number of variables and the number of equations, we need 2 equations. There is a high chance that C is the correct answer. Using 1) & 2), from a=1 and b=-1, 1, the answer is both yes or no. Hence, the conditions are not sufficient and the correct answer is E.
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ANS: There are 2 variables in the original condition. Hence, we need 2 equations, and there is a high chance that C is the correct answer. Using 1) & 2), from a=1 and b=-1, 1, the answer is both yes and no. The conditions are not sufficient and the correct answer is E.
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What is the perimeter of a certain right triangle?

1) The hypotenuse’s length is 5 2) The triangle’s area is 4.5

ANS: From the original condition, we can see that the number of variables is 2. Hence, the number of equations we need is 2 as well. The correct answer is C
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ANS: If we modify the original condition and the question, we get 2^(x+y)/2(x-y)=? --> 2^((x+y)-(x-y))=2^2y. We only have to know y and the correct answer is B.
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When a positive integer n is divided by 29, what is the remainder? 1) n-5 is divisible by 29 2) n-29 is divisible by 5

ANS: If we use direct substitution, in case of con 1), the remainder is always 5. The answer is unique and the condition is sufficient. Thus, the correct answer is A.
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If b is an integer greater than 1, ab=? 1) b^a=1 2) b=2

ANS: Since there are 2 variables in the original condition, there is a high chance that C is the correct answer. However, we have to apply the common mistake type 4(A) since it is an integer question, one of key questions. In case of con 1), since we have a=0, we get ab=0. The answer is unique and the condition is sufficient. Hence, the correct answer is A.
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If a, b are positive integers (b≠1), is a/b<1/10 ? 1) a/(b-1)<1/15 2) (a-1)/b<1/15

ANS: Since there are 2 variables in the original condition, there is a high chance that C is the correct answer. Applying the common mistake type 4(A), the correct answer is A.
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ANS: If we modify the original condition and the question, from (x-y)^2-(x+y)^2=? --> -4xy=? Hence, we only have to know the value of xy and the correct answer is A.
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*ANS: There are 2 variables in the original condition and there is a high chance that C is the correct answer. Using con 1) and con 2), we get u=v=0. Hence the correct answer is D.
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How many 5 existed in the 11 numbers? 1) The average (arithmetic mean) of the 11 numbers is 5 2) The median of the 11 numbers is 5

*ANS: If we modify the original condition and the question, since we have 11 numbers, the median exists in these numbers. Hence, if we look at the condition 2), since the median is 5, 5 is always a part of 11 numbers. The answer is yes and the answer is B.
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ANS: There are 2 variables (x and y) and there is a high chance that C is the correct answer. Using con 1) and 2), the answer is yes when xy=1 and the answer is no when x=y=-5. Hence, the condition is not sufficient. So, the correct answer is E.
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